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Are there examples of production functions where increasing the input of one factor and keeping the other factor constant leads to reductions in total production?

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Production functions are defined without specific values for parameters, so they all could if you impose that the logical parameter implies a negative return.

For example, consider a Cobb-Douglas production function of capital and labor,

$Y=\beta_0 K^{\beta_k}L^{\beta_l}\omega \varepsilon$

where $\omega$ denotes firm-observed productivity and $\varepsilon$ is an idiosyncratic shock. If you wanted to augment this with an input that descreases prodcution, maybe $P$, you'd just write,

$Y=\beta_0 K^{\beta_k}L^{\beta_l}P^{\beta_p}\omega \varepsilon$

and impose that $\beta_p<0$.

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    $\begingroup$ Thank you, that was easy. But then we might as well remove this factor from the production function because a rational firm would never choose $P>0$? I was thinking more of a setup where under certain combinations of factors or at certain levels of production negative total returns kick on. $\endgroup$
    – Papayapap
    Jul 11 at 16:20
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    $\begingroup$ I mean, if P is exogenously assigned due to government regulations (like safety inspections), it would be > 0. $\endgroup$ Jul 11 at 16:23
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    $\begingroup$ Most production functions exhibit decreasing returns to individual inputs, but are not in functional forms where it would be negative, just that returns approach 0. (e.g. Cobb-Douglas). $\endgroup$ Jul 11 at 16:25
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    $\begingroup$ Another reason why P might be positive is if it's a joint effect of the use of one of the factors of production. E.g. when the use of capital increases pollution (P) which has an effect on output $\endgroup$ Jul 12 at 9:24

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